Environmental Engineering Reference
In-Depth Information
NS = (0, 0,25, 0,5), S = (0,25, 0,5, 0,75), D =
(0.5, 0,75, 1).
NEARLY SAME, DIFFERENT). The method
uses the distance between the fuzzy diversity score,
represented by the fuzzy triangular number for
each strategy S
ij
, and each of the initial linguistic
terms to represent the degree to which
obtained
score,
is confirmed to each of them. For instance,
the distance between the obtained fuzzy diversity
score D
ij
and the expression same, nearly same,
different is defined as follows:
The aggregation stage:
During this stage all
linguistic values provided by experts are aggre-
gated to obtain a collective assessment for the
alternatives. It is provided by calculation of the
fuzzy diversity score D
ij
as an arithmetic mean:
D = (
1
t
w ×m ,
1
t
∑
t
∑
t
t
w ×
ij
k
ij
k
1
2
i=1
i=1
(13)
2
( )
(
)
(
)
,
1
t
r
∑
µ
3
d D , SAME =
j
-
j
;
t
µ
∑
t
t
α
w × )
β
ij
ij
Dij
same
j=1
ij
k
ij
i=1
1
2
Where w
k
- weight of k diversity criterion;
< m ,
3
2
( )
(
)
(
)
r
t
α β
a triangular fuzzy number that
represents one of linguistic values {S, NS, D}
assigned by
t
th expert for S
ij
diversity strategy.
D
ij
represents a distance between two objects:
primary and secondary RTS. The more distance
D
ij
, which corresponds certain diversity strategy
S
ij
, the more diverse both RTSs. In this case, the
primary RTS with its diversity attributes is con-
sidered as centre of cluster.
However, the final result is a fuzzy set, which
does not correspond to any label in the original
term set. In this case, “linguistic approximation”
is needed (Zadeh, 1999). The process of linguistic
approximation consists of finding a label, whose
meaning is the same or the closest (according
to some metric) to the meaning of an unlabeled
membership function generated by some compu-
tational model.
It is worth to note that results, obtained by the
fuzzy arithmetic, are fuzzy sets that usually do not
match any linguistic term in the initial term set,
so a linguistic approximation process is needed
to express the result in the original expression
domain.
Using the best-fit method (Dubois, et al.,
1980), the obtained fuzzy diversity score D
ij
for
each strategies S
ij
can be mapped back to one
(or all) of the defined linguistic terms (SAME,
t
t
∑
µ
j
j
,
> -
d D , NEARLY SAME =
-
µ
;
ij
ij
ij
ij
ij
Dij
NS
j=1
1
2
( )
(
)
(
)
2
3
r
∑
µ
j
j
d D , DIFFERENT =
-
µ
;
ij
ij
Dij
different
j=1
(14)
Hence, each S
ij
diversity strategy is character-
ized by 3-tuple
< d , d , d >
ij
(3)
, where,
d
i
(r)
a
distance between the obtained fuzzy diversity
score and the corresponding linguistic term
(SAME, NEARLY SAME, DIFFERENT).
It should be noted that each,
d
i
(r)
(j=1,…J,
where
j
- a number of possible alternatives clas-
sified as a type of S
i
strategy) is an unsealed
distance. The closer D
ij
, is to the
r
th expression,
the smaller
d
i
(r)
is. More specifically,
d
i
(r)
is equal
to zero if D
ij
, is just the same as the
r
th expression
in terms of the membership functions. In such a
case, D
ij
should not be evaluated to other expres-
sions at all due to the exclusiveness of these ex-
pressions. To embody such features, new indices
need to be defined based on
d
i
(r)
(r = 1, 2, 3).
Suppose
d
i
(3)
is the smallest among the obtained
distances for D
ij
, and let α
i1
, α
i2
, α
i3
represent the
reciprocals of the relative distances between the
(1)
(2)
ij
ij
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